A Local Epsilon Version of Reed's Conjecture

In 1998, Reed conjectured that every graph of maximum degree Δ and clique number ω can be colored with ⌈12(Δ+1+ω)⌉ colors, significantly strengthening Brooks' Theorem. As evidence for his conjecture, he proved that this is true instead when the number of colors is some nontrivial convex combination of Δ+1 and ω. I 1979, Erdős, Rubin, and Taylor proved that a connected graph G is L-colorable for every list-assignment L satisfying |L(v)|≥d(v) for all v∈V(G), unless every block of G is a clique or odd cycle. We ask if every graph G is L-colorable for every list-assignment L satisfying |L(v)|≥⌈12(d(v)+1+ω(v))⌉, where ω(v) denotes the size of the largest clique in G containing v. We prove that this is true instead when |L(v)| is some nontrivial convex combination of d(v)+1 and ω(v), under certain mild assumptions.